Defense Intelligence Reference Document An Introduction To The Statistical Drake Equation

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(26)
where (25) wa~ already used in thc la~t stcp. By
virtuc of the uniform prohahility density function
( 10) and of (26), the general tramformation law (9)
finally yields
b1 - a 1
(27)
In other words, the requested pdf of Y, i~
i = I. .... 7 lm(a,)sy,ln(h,)I (28)
Probability density functions of the natural logs of
all the u11ifarmly di.~tributed Drake randam
variable~· D, .
This is indeed a positive function of.\· over the
interval In(uJ :S: y :S: ln(hJ, as for every pdf, and it 1s
casy lu sec thal its normaliLation cundition is
fulfilled:
... (29)
Next we want to find the mean value and
standard deviation of Y, , since these play a crucial
role for fulurc dcvclopmcnl~. The mean ialuc (Y,) i~
given by
i
ln(I> ) ( ) iln(I,, I v . e ,.
(Y,)= 'v· f V dv= -·--d;
ln(u,l • Y, - • ln(u,)b;-(1; •
b, [In(b, )- I]- a, [1,;(u, )- I]
b1 - 11,
(30)
This is thus the mean value of the nat11ral log of all
the unifarmly di.~tributed Drake random variable.1·
D,
36
In order to find the variance abo, we must first
compute lhc mean value of the square of Y,, lhat is
= h, [in 2 (h 1 )- 2 ln (h 1 )+ 2 ]- a, [tn 2 (a,)- 2 ln(a; )+ 2]
h; -,1;
... (32)
The variance of Y; = 1n(D;) is now given by (32)
minus the square of (31 ), that. after a few reductions.
yield:
2 a,h;[ln(h,)-ln(a,)] 2
=a,n(n,l=l- ( )'
h, -a,
Whence the corresponding standard deviation
(34)
Let us now turn to another topic: the use of
fouricr tran~forms, that, in probahility theory, arc
called ·'characteristic functions," following again the
notations of Papoulis (ref [51) we call "'characteristic
function", y_ (;) , of an assigned probability
distribution Y1 , the Fourier transform of the relel'ant
probability density function, that is (with j = μ)
The use of characteristic function~ simplifies things
greatly. For instance, the calculation of all moments
of a known pdf becomes trivial if the relevant
characteristic function is known, and greatly
simplified also arc the proofs of important theorems
of slalistics, like lhc Ccnlral Limit Theorem thal wc
will use in Section 4. Anolhcr imporlanl rcsull i~ that
thc characteristic function of lhc sum of a finitc
numhcr of independent random variable~ is simply
gi vcn by the product of the corresponding
characteristic functions. This i~ just the case we arc
facing in the Statistical Drake equation (3) and so we
are now led to find the characteristic function of the
random l'ariable Y1. i.e.
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